> Given that, for R[0,1]:>> a) each irrational has a unique infinite expansion as path

That is the question. If so, why has never anybody written it usingdigits or bits?>> b) each initial segment of the expansion is the initial segment of a> rational>> c) every rational's path is in the tree

That is the question too. Why has never anybody written the completedecimal- or binary expansion of a periodic rational?>> d) the union of finite initial segments of the expansion as tree> contains the expansion as path>> e) thus each irrational's expansion is a path in the tree of rationals>> then, yes, that appears to be so.

I agree with your conclusion but not with the premises.

Remember: Never has anybody written an infinite sequence other than byusing the symbolic method: "1/9" or "1/pi" or "1/(SUM 1/n!)". Thesehowever are only names to identify or formulas to construct infinitepaths - not paths that belong to the Binary Tree.